The combinatorial Riemann mapping theorem

Acta Math., 173 (1994), 155-234

The combinatorial Riemann mapping theorem

by

JAMES W. CANNON(I)

Brigham Young University Provo, UT, U.S.A.

I. I n t r o d u c t i o n

The combinatorial Riemann mapping theorem is designed to supply a surface with local quasiconformal coordinates compatible with local combinatorial data. This theorem was discovered in an attempt to show that certain negatively curved groups have constant curvature. A potential application is t h a t of finding local coordinates on which a given group acts uniformly quasiconformally. The classical Riemann mapping theorem may also be viewed as supplying local coordinates (take a ring and map it conformally, by the classical theorem, onto a right circular cylinder; pull the resulting flat coordinates back to the ring as canonical local coordinates). This coordinatization role is disguised in the classical case by the fact that a Riemann surface comes preequipped with local coordinates in the desired conformal class. In the combinatorial case we begin with a topological surface having no presupplied quasiconformal structure and our task is t h a t of discovering the local coordinates (again by pulling coordinates back from an appropriate right circular cylinder).

The combinatorial d a t a are supplied by coverings of the surface called

shinglings.

A shingle

is a compact connected set. A

shingling

is a locally finite cover of the surface by shingles. (A shingling is like a tiling except t h a t shingles are allowed to overlap while tiles usually do not overlap.) A shingling may be viewed as a combinatorial approximation to the surface. A given shingling, being locally finite, gives only a first approximation to a local quasiconformal structure on the surface. The total structure can only be determined by a sequence of finer and finer shinglings. The problem becomes t h a t of (1) This research was supported in part by NSF research grants. We gratefully acknowledge fur- t h e r support by the University of Wisconsin--Madison, Brigham Young University, the University of Minnesota and the Minnesota Supercomputer Institute, the Geometry Supercomputer Project, and Princeton University during the period of this research.

12 - 945204 Acta Mathematica 173. lmprirnd le 2 ddcembre 1994

156 J . w . CANNON

determining when the approximate structures supplied by a sequence of shinglings are compatible so that there is a limiting quasiconformal structure. The method is t h a t of

extremal length.

Extremal length has been studied both in the continuous [LV] and in the discrete [D] settings. We will mesh the two settings by taking the limit of discrete conformal optimizations.

Our approach to the problem is set out in Section 2 where we describe the classical theorem as the solution to a certain variational problem. For each shingling there is a corresponding finite variational problem which obviously has a solution. The remainder of the paper is then devoted to showing that, under appropriate conditions, solutions to the finite problems converge in the nicest possible way to a "combinatorial" Riemann mapping. The spirit of the undertaking is like that of Rodin and Sullivan, Beardon and Stephenson (see [RS], [Rol], [Ro2], [BS]), with the difference that they assume t h a t an underlying conformal structure is given from the start.

Acknowledgments.

Matt Grayson helped in an early formulation of the axioms. C u r t McMullen pointed out the importance of the Lipschitz condition in the proof of Theo- rem 5.9 and Corollary 5.10 and showed how the mapping theorem could be used to show that certain sequences of shinglings are not conformal. Peter Doyle showed me a number of references on combinatorial extremal length ([D], [M], [W]). Mladen Bestvina showed me the averaging trick of the proof of Theorem 7.1 so important in the final step of the mapping theorem. David Wright, Stephen Humphries, and William Floyd listened to many versions of tentative lemmas and propositions. Walter Parry [Pa] developed a nice list of properties of optimal weight functions. William Thurston helped me understand the difficulties, still unresolved, in making the combinatorial Riemann mapping theo- rem into a working tool. I thank them, and others forgotten for the moment, for their assistance.

The ideas that go into the proof of the theorem are very classical. I learned them primarily from [A], [DS], [Go], and [LV] but also in the classical treatments, [Ri] and [Hi]. Learning the classical arguments was a pleasure which extended over years. In making the arguments combinatorial, the clear connection with the classical was lost. In combinatorial form, the arguments are essentially self-contained and elementary, though lengthy.

Because I have opted to include almost all details, it is clearly possible to shorten the treatment in the following ways. Almost all of the propositions involving the approximate distance functions di would be obvious for true distance functions; once we convince ourselves that the di behave in approximately the same way, we can skip most of those details. Many of the geometric arguments break up into cases having the same result; the reader can work through one case with the confidence t h a t the other cases work similarly.

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 157 Many steps of the numerical calculations are included; since they are essentially routine, they could be omitted. Skip what y o u will.

2. T h e R i e m a n n m a p p i n g a s a v a r i a t i o n a l p r o b l e m

T h e Riemann mapping theorem first appeared in [Ri]. T h e Riemann mapping theorem, in one of its versions, may be stated as follows. Suppose T~ is a closed topological annulus or ring in the complex plane. T h e n 7~ inherits a natural Riemannian metric ldz[ and a natural area form dx.dy. A conformal change of metric on 7~ multiplies the metric by a positive function Q--:~(z) and the area form by the positive function Q2,

RIEMANN MAPPING THEOREM. There is a positive continuous function Q such that the resulting Riemannian structure Q]dz[ on 7~ is metrically a right circular cylinder, say of height H, circumference C, and area A = H C .

It has always amazed me that Riemann could even have conjectured this theorem.

Apparently what happened was this. (See [Poi, Chapter 1].) Think of T~ as a uniform conducting metal plate. Apply a voltage, maintaining one of the b o u n d a r y curves at voltage H , the other at voltage 0. The current must flow and stabilize. T h e n the lines of equipotential form a family of simple closed curves filling up 7~ and separating the ends of T~. T h e current flow lines also fill up T~ and are arcs joining the ends of ~ . These two families of lines meet orthogonally, give flat coordinates to ~ , and turn ~ into a right circular cylinder. T h e ratio (H/C) may be thought of as the resistance of the ring as a conducting plate to current flow between the ends. It is a conformal invariant. It is called the analytic conformal modulus of the ring 7~. Note t h a t

H 2 A ( H / C ) = A = C -~"

T h e r e is a wonderful trick for creating conformal invariants. (See [A].) For a fixed Riemannian surface, one simply assigns a number to each metric conformally equivalent to the given one and then takes either the supremum or the infimum of those numbers over all of the metrics.

T h e resistance or modulus (H/C) is precisely such an invariant. (See [LV, Chap- ter 1].) It may be realized as follows. With each metric multiplier Q associate a ~-area A(Q), a ~-height H(~), and a Q-circumference C(~) which gives respectively the area, the minimal distance between the ends, and the minimal distance around the ring with respect to the new Riemannian metric ~. I dzl and the new area form Q2 dx.dy. T h e n we have

(H/C) = sup H(~)2 = inf A(Q) A(~) C(#) 2"

158 J . w . CANNON

Furthermore, b o t h the supremum and infimum are realized by t h a t positive multiplier function Q which turns 7Z into a right circular cylinder. T h a t is, the optimal function Q is the absolute value of the derivative of the Riemann mapping. This circumstance will play a central role in our combinatorial theorem where the lack of local coordinates makes the definition of derivatives, let alone their use, difficult.

We shall be dealing with surfaces for which no Riemannian structure is given. Our principal potential application is to 3-manifold groups. According to [F], IBM], and observation, such groups often have a visual topological 2-sphere at infinity which has no obvious preferred Pdemannian structure. The entire goal will be to find an appropriate Riemannian or quasiconformal structure on the surface on which the group of covering translations acts conformally. T h e group also often has a nice recursive combinatorial structure at infinity (see [C1], [C2], and [Gr]). This combinatorial structure at infinity creates shinglings at infinity which are of the type which we shall be studying in this paper. We lose no generality in assuming the existence of a topological metric on the surface. This metric will allow us to talk about the rough size of objects on the surface.

Let S be an arbitrary shingling of our topological surface. Then ,~ may be used to define an approximate metric and approximate area for subsets B of the surface. Simply define b o t h the length and area of B to be the number of elements of the shingling t h a t intersect B. T h a t is, assume that each element of ,~ has length and area equal to 1. It is then analogous to the classical case if we make a "conformal" change of approximate metric by changing the length of the element to 0 and the area of an element to Q2. T h e number Q may be an arbitrary nonnegative function on S. T h e 0-length and e-area of B are then simply the sums of the element lengths and areas for elements intersecting B.

If Tr is a ring in our surface, we obtain heights, circumferences, and areas H(Q), C(Q), and A(Q). Varying Q over all possibilities for which A(Q)#0, we obtain two approximate conformal moduli,

Msup(~,,5) = sup H(O)2 o A ( e ) and

mi~t(7~, S) = inf A(e) C(e) 2"

It is a fact which we shall prove elsewhere (Theorem 7.1) that, if the surface is Riemann- ian, if the elements of ,~ are fairly round, fairly small relative t o the size of ~ , and do not overlap too much, rather like a slightly expanded circle packing, then the approximate conformal moduli will fairly closely approximate the analytic conformal modulus of ~ .

W h a t we have argued is that every shingling gives an approximate notion of confor- mal modulus to every ring in the surface. Now we pass to a sequence of such approxi- mations.

THE COMBINATORIAL RIEMANN MAPPING THEOREM 159 Fix K = K ( 1 ) > 0 . (We use the modifier, (1), because we shall actually study a whole sequence of constants associated with a sequence of shinglings, of which K is the first constant on which the others will depend.) A K-interval is a real interval of the form [r, K.r], r > 0 . Let 81,82, ... denote a sequence of shinglings of a topological surface with mesh (largest element size) locally approaching 0. This property that mesh locally approaches 0 is independent of the particular topological metric with which the surface is endowed. We say that this sequence is a conformal sequence (K) if:

(i) for each ring ~ , the approximate moduli Msup(~'~,8i) and minf(T~,8i), for all i sufficiently large, lie in a single K-interval [r, K.r]; and

(ii) given a point x in the surface, a neighborhood N of x, and an integer I, there is a ring T~ in N \ ( x } separating x from the complement of N, such that for all large i the approximate moduli of ~ are all greater t h a n I.

The intuitive content of these conditions is as follows. The first condition says that the approximate moduli defined by the shinglings are well-defined, at least asymptotically and up to multiplication by a uniform constant. This is a very strong condition that is difficult to recognize and generally depends on having the elements of the shinglings of uniform shape and placement. The multiplicative constant is unavoidable as an artifact of the combinatorial approximation. The second condition is the combinatorial analogue of the geometric property of a planar surface which says that around each point there is an arbitrarily small circular ring such t h a t the ratio of the outer radius to the inner radius is arbitrarily large. This condition keeps points from exploding in the limit.

If approximate moduli are to approximate classical moduli in any sense, then con- ditions (i) and (ii) are clearly necessary. The combinatorial Riemann mapping theorem says t h a t the conditions are also sufficient.

COMBINATORIAL RIEMANN MAPPING THEOREM. L e t 3 1 , 8 2 , ... be a conformal s e - q u e n c e K(1) of shinglings of a (metric) topological surface. Then there is a constant K ' = K ' ( K ( 1 ) ) satisfying the following condition. If T~ is any ring in the surface, then there is a metric D on 7~ which makes T~, isometrically, a right circular cylinder and in which classical moduli and asymptotic approximate moduli are K'-comparable. That is, if T~' is any ring in T~, then there are an integer I and a Kr-interval It, Kr.r] such that the classical D-analytic modulus of 7~ ~ in T~ and the approximate moduli Msup(T~,Si) and minf(~', 8i) all lie in [r, K'.r] for each i>/I.

The content of the theorem is the existence of local analytic coordinates for which classical modulus is approximated by asymptotic combinatorial modulus.

COROLLARY. T h e r e exists a quasiconformal structure on the surface whose analytic moduli are uniformly approximated by asymptotic combinatorial moduli. (Note that the

160 J . w . CANNON

quasiconformal structure is unique.)

Proof of the corollary. Cover the surface by rings R 1 , T ~ 2 , . . . . Let D1,D2,... be metrics as promised by the theorem. We need to show that the transition functions

(hi n 7z~, Di)

^-*

(n~ nnj, Dj)

are uniformly quasiconformal. Let R ' be a ring in

n~nnj.

Let [r~, K'r~] and [rj, K'rj] be K ~ intervals and N a positive integer such that for n > N , the Di-modulus Msup(~ ~, Di) and the approximate modulus M~up(n',Sn) lie in [rn, K'r~], while Msup(n', D j ) and M~up(T~ t, Sn) lie in [rj, K'rj]. Then, if r = m i n ( r i , rj), the set

{Msup(R', Di), M~up(g', Dj)}

lies in [r, K'2r]. By [LV, T h e o r e m 7.2, p. 39], we conclude that the transition function is

K'2-quasiconformal. []

The remainder of the paper is devoted to the proof of the combinatorial Riemann mapping theorem. We fix therefore a conformal sequence (K(1))

~ 1 , ~ 2 , ...

with mesh going locally to 0 and a ring R. Our goal is to endow T~ with a special metric D which makes T~ a right circular cylinder and whose analytic moduli are in appropriate range.

3. Optimal weight functions

For each shingling $ of a ring R we shall prove (Proposition 3.1) the existence of a weight function ~: 8 - * [0, co) which is optimal in the sense that it realizes the supremum in the definition of Msup(R,S). We can therefore associate with the shinglings 81,82, ... (of our conformal sequence) a sequence of optimal weight functions ~1, ~2, .... Of course, with any weight function Q there are associated length and area functions L(O) and A(Q). Therefore we obtain approximate length functions L1,L2, ..., and approximate area functions A1, A2, .... While useful, the length functions L1, L2, ... have potential defects that we have not been able to rule out. We therefore use a modified version of the functions Li which give us approximate distance functions dl, d2, .... The idea is to show that there is a subsequence such t h a t the approximate distance functions and approximate area functions converge, respectively, to a true metric d on R (see Section 4) and a well-behaved area function A on R (see Section 5). The metric d and the area

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 161 function A are then u s e d to define flat coordinates on 7r as follows. T h e vertical or y-coordinate which measures the distance from the ends of T~ is simply the d-distance from one end of 7r The horizontal or x-coordinate is then the derivative with respect to y of the area function. (See Sections 5 and 6.)

Consider t h e classical Riemann mapping theorem as discussed in Section 2. T h e r e is no a priori reason for assuming the existence of a nonzero weight function ~: 7~--* [0, co) realizing the supremum

H(c~) 2

H(Q) 2

A(a)

-- sup A(Q)

The famous gap in Riemann's original proof (see [Ri] and [Hi]) lay essentially in his assumption that such a function existed (the Dirichlet principle). However, the corre- sponding problem for shinglings is easy.

3.1. PROPOSITION (The existence of optimal weight functions).

There is a function

~=Qi:S~--~[O, 1]

such that

H ( a ) 2

H(O) 2

s u p

A(a) Q A(Q)

If a=9~ is normalized so that the associated area of the ring is 1, then we call gi an

optimal weight function

for (T~, Si).

Proof.

Pick a sequence 6(1), 6(2), ... of weight functions on

Si

such that

A(Q(j))#O

and such that

H(Q(j)) 2

A(Q(j))

converges to Msup(n, S~). Normalize each ~(j) by scaling

(H2/A

is scale invariant) so that A(p(j)) = 1. Then, if s ESi intersects 7~, p(j) (s) e [0, 1]. Hence, passing to a subsequence if necessary, we may assume t h a t

Q(j)(s)

converges, say to a ( s ) e [ 0 , 1]. If sC,~i and sNT~=O, define a ( s ) = 0 . T h e n A ( a ) = l and

H(a) 2 - H(a)2

Msup(T~, Si)

A(a)

as desired. []

There are of course optimal weight functions for the modulus minf(T~, $i) as well.

T h o u g h we use such optimal weight functions only indirectly in this paper, it would be necessary to study them in determining that the axioms are satisfied by specific sequences of shinglings.

Optimal weight functions are fascinating. We will summarize here some of their abstract properties as developed by Walter P a r r y ([Pa D. Let S denote an abstract finite set. It corresponds to the shingling above. Let V denote a real vector space with 8 as

162 ~. w. CANNON

basis. T h e n every element of V may be considered to be an S-tuple of real numbers.

We may think of the square of the length of an element of V as the area of t h a t vector.

An abstract p a t h in V is a nonzero vector each coordinate of which is 0 or 1. Another way of viewing an abstract p a t h is as a subset of 8; an element of S is in the p a t h if its coordinate is 1. Let P denote a distinguished collection of paths. A weight vector w on S is a unit vector in V each coordinate of which is nonnegative. In terms of area, we are considering only those weight vectors of area 1. Define the height

H(w, P)

of S with respect to w and P to be the minimal inner product of w with the elements of P . A weight vector w is said to be optimal (with respect to P ) if, among all weight vectors,

H(w, p)2

is maximal.

Note t h a t to maximize

H(w,

p ) 2 is to minimize 1

H(w, p)2"

Hence, if we had called

H(w)

the circumference instead of the height, our single opti- mization would have included b o t h of the optimizations considered in defining conformal moduli and approximate moduli above.

We prove here only the first of P a r r y ' s theorems, namely t h a t there is a unique optimal weight vector. We will then summarize his other results.

3.2. PROPOSITION

([Pa D. There i8 a unique optimal weight vector.

Proof.

Let w and w' be distinct nonnegative unit vectors such t h a t

H(w)>~H(w').

If rE(0, 1), then set

v--tw+(1-t)w'.

It is enough to show t h a t

Let

pEP.

T h e n

[iv[ I

v.p= [t(w.p)+(1-t)(w'.p)]

~ [ t H ( w ) +

( 1 - t ) H ( w ' ) ]

~vHg(w') > H(w').

^[]

We define a weight vector to be a

flow vector

if it is in the nonnegative cone of the paths. Intuitively it is then a bundle of paths. Every weight vector w determines a subset of paths, namely those paths p E P such that

w.p= H(w, P).

We call those paths

minimal

for w. We call w a

current flow vector

if it is in the nonnegative cone of its minimal

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 163 paths. P a r r y characterizes the unique optimal weight vector as the unique current flow vector. He proves t h a t some positive scalar multiple of this vector has integer entries.

In our situation, there are two obvious choices for the collection P of paths. For the first, call a subset of S an abstract path which joins the ends of T~ if it is the collection of elements of S which intersect a topological p a t h in T~ joining the ends of 7~. It is with respect to this collection of abstract paths t h a t we have defined height. For the second collection, say that a topological simple closed curve circles T~ if it separates the ends of ~ . Call a subset of S an abstract p a t h circling T~ if it is the collection of elements of S which intersect a topological simple closed curve in 7~ circling ~ . It is with respect to this collection of abstract paths that we have defined circumference. Our two optimization problems then become that of finding a current flow vector joining the ends of ~ and a current flow vector circling ~ .

P a r r y proves one more result, namely t h a t since, in our situation, each p a t h which circles T~ intersects each p a t h which joins the ends of T~, minf(~,S)~<Msup(T~, S). The axioms for a conformal sequence then deal with the further relationship of minf(~.,Si) to M s u p ( ~ , 3 j ) as T~, i, and j vary.

We will use P a r r y ' s results in discussing examples but in no other way.

We now leave the discussion of optimal weight functions in the abstract and take Q1, Q2,-.- to be the unique optimal weight functions associated with our conformal se- quence 81,$2, .... In addition to the optimal weight functions Q1, Q2, ..., we have the associated length and area functions L1, L2, ... and A1, A2, .... We also obtain height functions /-/1,//2, ... and circumference functions C1, C2, ... for ~ and other rings con- tained in 7~. Much of the remainder of the paper will be devoted to the limiting properties of the functions Li and Ai.

Actually, the length functions L1, L2, ... are difficult to work with since there is no a priori reason for assuming that points of 7~ have short Li-length. We resolve this difficulty by following the admonition of G. Polya's "traditional mathematics professor"

[Pol, p. 208]: "My m e t h o d to overcome a difficulty is to go around it." We show that, though a point may have large length, it has nice neighborhoods with short frontiers in 7~. This allows us to avoid and ignore points of long length. Our nice neighborhoods will be called proper disks. (See Figure 1.) A disk DCT~ is proper if it is closed and either lies in Int ~ or intersects Bd 7~ along a b o u n d a r y arc. T h e frontier of D in ~ , denoted F r D , is either an arc a properly embedded in ~ or a simple closed curve ~ in Int :R. T h e relative interior D \ F r D is denoted by ~ Int D. T h e content of the next proposition is t h a t each point has proper disk neighborhoods with short frontiers.

3.3. PROPOSITION (Existence of proper disk neighborhoods with short frontiers).

Given x E ~ , a neighborhood N of x, and e > 0 , there is an annulus T~(x,N,s) in N

164 J. W. CANNON

Fig. 1

separating x from the complement of N, and an integer I(x, N, ~) such that, for each i>~I(x,N,~), there is a proper disk neighborhood D = D ( x , N , c , i ) of x in N such that FrD lies in T~(x,N,~) and such that Li(FrD)<~. (See Figure 2.)

Proof. We may assume that NMT~ is a proper disk D(N). By property (ii) in the definition of conformal sequence, there is an annulus T~(x, N, ~) in N separating x from the complement of N and an integer I ( x , N , c ) such that, for each i ~ I ( x , N , c ) , the approximate modulus

minf(T~(x,

N, E), Si) is greater than 1/~ 2. Let J be a simple closed curve circling T~(x, N, c) of minimal Li-length in general position with respect to Bd 7~.

Let J~ be a component of JMT~ separating x from Fr D(N). Let D be the proper disk neighborhood of x with F r D = J ~. It remains to see that J, and hence F r D = J ~, has Li-length less than ~. If Li(J)=O, we are done. Otherwise,

Ai(7~(x,N,E)) 1

LC 2 < minf(•(X, N, ~),

8i) < Ci(~(x,

N, ~))2

<" Li(J)---~"

T h a t is, Li(J)<e. []

Having proved the existence of proper disk neighborhoods with short frontiers, we first pass to a subsequence of the Qi's in order to ensure that these neighborhoods cover T~ uniformly. We then use these neighborhoods to define modified approximate distance functions di on T~ which behave b e t t e r t h a n the length functions Li.

We shall use in what follows the notion of the star of a set in a collection. Let X be a set and Y a subset. Let C be a collection of subsets of X . The star star(Y, C) of Y in C is the union of the elements of C that intersect Y. T h e star operator may of course be iterated so that we have sets starn(Y, C ) C X for all n > 0 .

By Proposition 3.3, we may assume after passing to a subsequence that the follow- ing condition is satisfied. For each x CT~ and for each i C Z+, there exists an annulus 7~(x,i) of metric diameter < 1 / i surrounding x having the following property. If j>~i, then T~(x, i) misses star2(x, Si) and there is a proper-disk neighborhood D = D ( x , i , j ) of nMstar2(x, Sj) whose frontier F r n lies in T~(x,i) and has length i j ( F r D ) < l / i .

THE COMBINATORIAL RIEMANN MAPPING THEOREM 165

Fig. 2

We associate with

Li

a modified a p p r o x i m a t e distance function

di:

R xT~---~[0, o~) as follows. Let xET~. An

i-approximation

to x in T~ is a proper disk

D(x)

of metric diameter <

1/i

such t h a t 7~Nstar2(x, Si) cT~ Int

D(x)

and such t h a t

Li(Fr D(x)) < 1/i.

By our choices in the previous paragraph, every xET~ has an /-approximation. An i-

approximate path

from x to y is a p a t h in 7~ j o i n i n g / - a p p r o x i m a t i o n s of x a n d y. T h e

approximate distance di(x, y)

from x to y is the m i n i m u m Li-length of a l l / - a p p r o x i m a t e p a t h s from x to y in 7~.

Since we shall have m a n y propositions t h a t deal with the functions

di,

we take a m o m e n t to mention two fundamental techniques used in dealing with them.

T h e first technique is this. Suppose t h a t / - a p p r o x i m a t i o n s

D(x)

and

D(y)

intersect but t h a t their frontiers

FrD(x)

and

FrD(y)

do not. T h e n one of the two sets contains the other, say

D(x)cO(y).

T h e n

D(y)

is a n / - a p p r o x i m a t i o n to x. Any arc joining

D(x)

to something also joins

D(y)

to t h a t same set. Thus we m a y in almost all cases replace

D(x)

by

D(y)

as a n / - a p p r o x i m a t i o n to x. But then

FrO(x)

and

FrD(y)

do intersect, in fact coincide. In s u m m a r y of the first technique,

if two i-approximations intersect, we may assume that their frontiers intersect.

T h e second technique considers the nonadditivity of length; an arc a which is the concatenation of arcs a0 and a l will in general not have Li-length t h a t is the sum of the Li-lengths of the subarcs. In fact, the length of a m a y equal the length of one of the subarcs. T h e p r o b l e m is t h a t there m a y be a shingle which hits b o t h subarcs. T h e weight of this shingle will be counted twice in the s u m of the lengths, only once in the length of the sum. One can avoid this problem in developing lower bounds for the Li-length of s by taking subarcs s0 and Sl of s which intersect no c o m m o n shingle. We d e m o n s t r a t e the second technique by example. Let ~ 0 and 7~1 denote the ends of T~. Let J denote a simple closed curve in the interior of 7~ separating the ends of 7~. Let s join the ends of ~ . Let s0 denote an open subarc of s whose closure is irreducible from star(7~0,3i)

166 J . W . CANNON

to star(J, Si). We say that So is

open-irreducible

from star(7~0, Si) to star(J, Si). Let cq be open-irreducible from

star(E1, Si) to star(J, 8/).

Then, clearly, no shingle hits b o t h c~0 and a t so that

Li(c~) >1 ii(~o)T ii(oq ).

How long are C~o and a l ? Let P0 denote the endpoint of C~o in s t a r ( R o , S i ) , pl the endpoint in star(J,

Si).

Let so denote a shingle which contains Po and hits 7~o. Let st denote a shingle which contains Pl and hits J. Let qo denote a point of 7~o in so. Let ql denote a point of J in sl. Let

D(qj)

denote an i-approximation to qj, j = 0 , 1. Then

pj

lies in the relative interior of

D(q3)

so t h a t ao is an i-approximate arc from 7~o to J.

In particular the di-distance from 7~ to d is a lower bound on the Li-length of ao.

In order to examine the Li-distance between the same two sets, we need to consider two cases. If the two i-approximations intersect, then by technique one we may assume that their frontiers intersect. In that case, the union of their frontiers joins 7~o to J so that the Li-distance from 7~0 to J is at most

2/i

and the di-distance is at most

1/i.

If on the other hand the two i-approximations are disjoint, then a0 hits b o t h frontiers, the Li-distance from 7~0 to J is therefore at most

Li(ao)+2/i. In either case, Li((~o) is at least as large as the Li-distance from 7~o to J minus 2/i. Note that the argument would have remained unchanged had we used

star 2

instead of

star. Technique two deals with such arguments involving open-irreducible arcs and their lengths.

T h e next proposition observes t h a t the function

di

repairs the apparent defect in the length function

Li

and, in the limit, satisfies the triangle inequality.

3.4. PROPOSITION (The functions

di

and the triangle inequality).

For each xET~, di(x,x)<l/i. For each x,y, zET~,

di(x, z) <~ di(x, y)+di(y, z)+ 2/i.

Proof.

The inequality

di(x, x) < 1/i

is immediate since we may take an arbitrary i-approximation

D(x)

to x and take an i- approximate p a t h from

D(x)

to itself in Fr

D(x).

As for the other, pick/-approximations D ( x ) ,

D(y), E(y),

and

E(z)

and paths c~ and ~, c~ joining

D(x)

and

D(y)

w i t h / i ( c ~ ) =

di(x, y),

and f~ joining

E(x) and E(y)

with

Li(~)--di(y, z).

By technique one above, we may assume t h a t if any two of these i-approximations intersect, so also do their frontiers.

It follows that the set

~UPr D(y)UFr E(y)Ut3

contains an i-approximate p a t h from x to z, and t h i s / - a p p r o x i m a t e p a t h necessarily has

Li-length

<. Li((~)+ 2/i + Li(~). []

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 167 3.5.

then

PROPOSITION. If (~ is an Li-minimal arc joining the ends T~o and 7~1 of T~, Li(aNstar(7~0,3i)) < 1/i.

Proof. Let X=(~nstar(T~0, Si). Let Y be a subarc of (~ open-irreducible from ~ 1 to star2(T~0, Si). Note that no shingle hits b o t h X and Y. Hence L i ( X ) < L i ( ( ~ ) - L i ( Y ) . By the argument of technique two above, L i ( Y ) > L i ( c ~ ) - 1 / i since c~ is Li-minimal joining

the ends of T~. Hence L ~ ( X ) < I / i as claimed. []

4. A p p r o x i m a t e distances and their limits

T h e aim of this section is to show that some subsequence of the approximate distance functions di converges uniformly to a limit d and t h a t the limit d is a true metric on 7~

compatible with the topological metric with which we began.

4.0.1. THEOREM (Existence of limit metric). Some subsequence of the approximate metrics dl, d2, ... converges to a true topological metric d on 7~ that is compatible with the given topology on Tt.

T h e fiat metric D whose existence is asserted by the combinatorial Riemann mapping theorem which makes ~ into a right circular cylinder will be a modification of d which takes into account not only the limit of the d~-distances (the ~i-lengths Li) but also the limit of the Qi-areas Ai. (After all, the optimal weight function takes b o t h lengths and areas into account.) While d alone exhibits strange local and global irregularities, the mesh of length and area will be completely well-behaved, a miracle which deserves to be b e t t e r understood.

T h e o r e m 4.0.1 is a consequence of two propositions, Proposition 4.0.2 and Propo- sition 4.0.3. We need a classical theorem before we can appreciate Proposition 4.0.2.

Functions f l , f2, ..- from a space X into a metric space (Y, d) are asymptotically equicon- tinuous if, for each x E X and for each e > 0 , there exist a neighborhood N of x and an integer I such that, for each i >~I and for each y, y' Eli(N), d(y, y')<~. T h e functions are uniformly bounded if all of the images lie in a single compact subspace of Y.

ARZELA-ASCOLI THEOREM. Suppose X is separable and (Y, d) metric. If f l , f 2 , . . . : X - - * Y

is uniformly bounded and asymptotically equicontinuous, then some subsequence con- verges to a function f: X - + Y . The limit function f is continuous. If the space X is compact, then the convergence is uniform.

168 J . w . CANNON

Proof.

Let C, compact in Y, contain the union of the images. Let x i , x 2 , ... be a countable dense set in X . Passing first to a subsequence of f ' s , we may assume that, for each i, the sequence f l ( X i ) , f 2 ( x i ) , ... converges to a point

f(xi)EC.

We claim that

fl(x),

f2(x), ... now converges for every

x G X .

Indeed, since

some subsequence

fl(x),

... e c ,

fi 1 (z),

^{. . .}

converges, say to

yEC.

Let ~>0. B y equicontinuity, there exist a neighborhood N of x and an integer I such that, for each

x ' E N

and for each

i>>.I, d(fi(x), fi(x'))<~.

Choose

x j E N .

Choose

K>~I

so large t h a t

k>~K

implies

d(fk(xj), f(xj))<c.

Choose

L>~K

so large that

l>~L

implies

d(f~(L)(x),y)<e.

Then, for

l>~L

we find

d(ft(x), y) <. d(fl(x), ft(xj))+d(fl(xj), f(xj))

+d(f(xj), A(t)(xj))+d(h(t)(xj), fi(t)(x))+d(fi(t)(x), y) < 5c.

Hence

fl(x), f2(x),..,

converges to y and we define

f ( x ) = y .

Thus

fl,f~,..,

converges to f . Finally we note that f is continuous at x. Indeed, if

x ' E N

and

i>~I,

d(f(x), f(x') ) < d(f(x), f,(x) f,(x') f(z') ).

T h e middle term is less t h a n e, the other terms approach 0 for i ~ o o . Uniformity of

convergence is checked similarly for X compact. []

4.0.2. PROPOSITION.

The sequence dl,

d2, ... : 7~ • 7~--* [0, oc)

is uniformly bounded and asymptotically equicontinuous.

COROLLARY.

Some subsequence old1, d2, ... converges to a limit function d: ~ x T~--*

[0, r

The function d is a continuous pseudometric.

Proof of the corollary.

Convergence to a continuous function d: R • oo) is a consequence of the Arzela-Ascoli theorem. T h e triangle inequality for d is a consequence of Proposition 3.4. S y m m e t r y

d(x, y) =d(y, x)

follows from the s y m m e t r y of each

di. []

4.0.3. PROPOSITION.

The limit pseudometric d separates closed sets X and points x in 7~ in the sense that, i f x ~ X , then d(x,X)>O.

Proof of Theorem

4.0.1. Proposition 4.0.2 and its corollary supply a subsequence of dl, d2, ... converging to a continuous pseudometric d. But a continuous pseudometric d satisfies the conclusion of Proposition 4.0.3 if and only if d is a true topological metric

on T~ compatible with the given topology on 7~. []

THE COMBINATORIAL RIEMANN MAPPING THEOREM 169 It remains to prove Propositions 4.0.2 and 4.0.3. The proofs appear in three sub- sections. Subsection 4.1 proves Proposition 4.0.2. Subsection 4.2 establishes a very important length-area inequality. Subsection 4.3 employs this inequality in the proof of Proposition 4.0.3.

4.1. U n i f o r m b o u n d e d n e s s and a s y m p t o t i c equicontinuity o f d l , d 2 , . . . (a p r o o f o f P r o p o s i t i o n 4.0.2)

4.1.1. PROPOSITION. The numbers H/(T~) and C/(T~) are uniformly bounded.

Proof. Choose an interval [r,K(1)r] and an integer I such that, for all i>~I, the moduli mi,f(n, S/) and Msup(7~, Si) lie in Jr, K(1)r]. Then for all weight functions Q on S~ yielding A(T~, Q)•0, we have

r ~< m i n f ( ' ~ , 84) <~ A(n,j_))

c(n, Q)=

and

H(T~, Q)2 <~ Msup(T~, 3/) ~< K(1)r.

A(7~, p)

Hence C(T~, Q)2-~<A(T~, Q)/r and H(7~, Q)2~<A(T~, Q).K(1)r. Since A(T~, ~ ) = A i ( ~ ) = I ,

the result follows. []

4.1.2. PROPOSITION. If Q is an optimal weight function for (T~,8) and if s E 8 has positive Q-weight, then there is an L(Q)-minimal path joining the ends of 7~ which intersects the shingle s.

Proof. Suppose the contrary. Let L be the minimum Q-length of a path through s joining the ends of T~. Redefine the weight of s to be Q(s). ( l - A ) w h e r e

0 < Q(s).)~ < L - H ( Q ) .

Let H ' and A t be the new height and area of 7~. By the optimality of Q, (H') 2 H 2

A

---7- ~< A " (1)

But direct calculation shows

H ' = H (2)

and

A' = A - 2AQ(s) 2 + A2 Q(s)2. (3)

170 J. W . C A N N O N

Substitution of (2) and (3) in (1) shows

-2 e(s)2+ 2e(s) 2 t> 0 (4)

which is absurd for A sufficiently near 0. []

T h e proof of this proposition is a model which will be improved and used a number of times before we axe through.

4.1.3. PROPOSITION (Every point is

di-near

to a point of positive weight).

IfxET~, then there exist a point x' ET~ of length

L~(x')>0,

i-approximations D(x) to x, and D(x') to x', and a path a joining Fr D(x) to Fr D(x') such that Li(~UFr D(x'))<l/i.

Proof.

If at any point we find

x', D(x), D(x')

such t h a t

FrD(x)NFrD(x')r

we are done since we may take

acFrD(x)NFrD(x')

and

Li(aUFrD(x'))=Li(FrD(x'))<l/i.

If possible, choose x' of positive weight in s o m e / - a p p r o x i m a t i o n to x. T h e n choose i-approximations D and E to x and x' respectively such that D and E intersect. By fundamental technique one, we may assume t h a t

FrDnFrE~O;

and we are done.

If no i-approximation to x contains a point of positive weight, then choose

D(x)

arbitrarily and let f~ be an arc irreducible from

D(x)

to the points of positive weight. Let x' be the terminal endpoint of ~ and

D(x')

an i-approximation to x'. If

D(x)ND(x')=r

then a s u b a r c a of f~ irreducible from Fr

D(x)

to Fr

D(x')

has Qi-lenth 0; and we are done.

Otherwise, since we can have neither

D(x)CD(x')

nor

D(x')cD(x),

Fr D(x)NFr D(x') # 0.

^[]

4.1.4. PROPOSITION.

The approximate distance functions di are uniformly bounded.

Proof.

Let xET~. Choose

x', D(x), D(x'),

and a ( x ) as in Proposition 4.1.3. Use Proposition 4.1.2 to choose a p a t h f~(x) which joins the ends of ~ , has Qi-length H(Qi), and intersects

FrD(x').

Given yET~, choose

y', D(y), D(y'), a(y),

and/~(y) similarly.

Let J be a simple closed curve of length

Ci

circling 7~. T h e n the union of the sets a ( x ) ,

FrD(x'),

f~(x),

J, /3(y), FrD(y'),

and a ( y ) is connected, joins

D(x)

to

D(y),

and has Li-length equal to or less t h a n

2/i+2H~+Ci.

From Proposition 4.1.1 it follows that the

numbers

di(x, y) are

uniformly bounded. []

4.1.5. PROPOSITION.

Let D denote a proper disk with Li(Fr D)<~e. Let a denote an Li-minimal path joining the ends of 7~. Then Li(aND) <~ + 2/i.

Proof.

Note that

Li(a)=Hi.

We may assume t h a t a hits D. Let ~ 0 and 7~1 denote the ends of T~. If T~0 misses star D, then let s0 be a subarc of a open irreducible from

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 1 7 1

7~o to star D. Otherwise, let a0 be empty. Define a t similarly. Since no shingle hits both a n D and the union a0Ual,

Li((~nD) <. H i - L i ( a o U a l ) . By the second fundamental technique, Hi is no longer than

ni(aoUal)+ Li(Fr D)+ 2/i.

Consequently,

Li((~nD) < Li(Fr D)+ 2/i <~ ~ + 2/i. []

4.1.6. PROPOSITION. Let D denote a proper disk in 7~ with Li(Fr D)<e. Assume every i-approximation to x in T~ lies in 7~ Int D. Then there is an i-approximation D(x) to x in T~ such that Fr D and Fr D(x) can be joined by an arc of Li-length <~+3/i.

Proof. By Proposition 4.1.3 there exist a point x' of positive Li4ength,/-approxi- mations D(x) to x and D(x') to x', and an arc a joining Fr D(x) to Fr D(x'),

Li(aUFr D(x') ) < 1/i.

By Proposition 4.1.2 there is an Li-minimal path/3 joining the ends of 7~ which intersects FrD(x'). Since D(x)C7~ Int D, the connected set

aUFr D(x')U(13RD)

joins FrD(x) to FrD. But by Proposition 4.1.5

Li[(aUFr D(x') )U(3nD)] < 1/i +(s + 2/i). []

4.1.7. PROPOSITION. The approximate metrics dl,d2, ... are asymptotically equi- continuous.

Proof. Given ( x , y ) e T ~ x ~ and e>0, we must find a neighborhood M x N of (x,y) and an integer J such that, for each ( x ' , y ' ) E M x N and for each i>~J,

IdJx, y)-di(x', Y')I <

Pick I so large that lO/I<e. If xTty, then require that the 1/I-neighborhoods of x and y be disjoint. As noted in the paragraphs following the proof of Proposition 3.3, there exist annuli, T~(x, I) and 7~(y, I), in the 1/I-neighborhoods of x and y having this property: if

13 - 945204 Acta Mathematica 173. Impfim6 le 2 d~cembre 1994

172 J . w . CANNON

i>~I,

there are proper-disk neighborhoods

D(x)=D(x, I, i)

of 7~Mstar2(x, $ ~ ) a n d D ( y ) =

D(y, I, i)

of 7~Mstar2(y,

Si)

such that Fr

O(x)CT~(x, I), Fr D(y)CT~(y, I), ni(Fr O(x))<

1/I,

and

ni(FrO(y))<l/I.

Pick connected neighborhoods M of x and N of y in 7~ and pick

J > I

so large that for each if> J, all/-approximations to all points of M miss T~(x, I) and all/-approximations to all points of N miss

R(y, I).

If

x=y,

then choose

N = M .

Take

x'EMMT~,

y'ENMT~,

i>~J

( > I ) . Then we have proper-disk neighborhoods

D(x)

o f x and

D(y)

of y as above with

ni (Fr D(x))

< 1 / I and

Li(Fr D(y))

< 1 / I . Take/-approximations

E(x')

to x' and

E(y')

to y'. T h e n

E(x')CT~

Int O(x) and

E(y')cT~

I n t O ( y ) . If

O(x)

and

D(y)

are disjoint, then we may argue as follows. Any p a t h joining

E(x')

to

E(y')

has a subpath a joining Fr

E(x')

to Fr

D(x),

a subpath ~ joining Fr

D(x)

to Fr

O(y),

and a subpath ~f joining Fr

D(y)

to Fr

E(y').

Let 5 denote an L~-minimal path joining

D(x)

to

D(y).

It follows from Proposition 4.1.6 that

Li(~) <. di(x',

y') < L i ( $ ) + 2 . [ 1 / I + 3 / i ] + 2 / I

since a and 7 can be chosen of lengths <

1/I+3/i.

On the other hand, if

D(x)

and

D(y)

do intersect, then one finds similarly that

0 < di(x', y') < 2. [1/I+3/i]+2/I.

In either case, the values for

di(x I, yl)

are restricted to a real interval of width less than

4/I+6/i<

1 0 / I < c . We conclude t h a t dl, d2, ... is asymptotically equicontinuous. []

Propositions 4.1.4 and 4.1.7 prove Proposition 4.0.2. []

4.2. An area-length i n e q u a l i t y

T h e following inequality will be useful in all t h a t follows. It should be viewed as saying that the approximate metrics

dl,d2,..,

have curvature ~>0. For each x E ~ , r > 0 , and i E Z+, we define

D(x, r, i) = {y e 7r y) <.

r}.

When x and i are fixed, we use the shorthand notation,

D(r)=D(x,r,i).

4.2.1. THEOREM (Quadratic estimate on area).

There is a positive constant

K(2)

such that, for each x C T~, each r > O, and each i E Z+ sufficiently large,

Ai[D(x, r,

i)] ~< K ( 2 ) . r 2.

THE COMBINATORIAL RIEMANN MAPPING THEOREM 173

Xl ~ 1

E ( z l )

^{E ( x )}

~ D ( x )

D ( Xo ) ~ ~ T~o

xo

Fig. 3

Remark

1. In any Riemannian ( = locally Euclidean) geometry defined on T~, this inequality would be satisfied a u t o m a t i c a l l y - - i n the small with K(2)~Tr since the geom- etry is locally Euclidean, in the large because the area is bounded. But in hyperbolic space the inequality fails for large r, and a scaling of the hyperbolic metric to create large negative curvature forces the inequality to fail for small r.

Remark

2. T h e argument for this theorem is our first really serious variational ar- gument using the optimality of the weight functions ~i. T h e arguments are of standard calculus of variations type. One modifies the given weight function, observes the conse- quence of optimality, and takes a limit as the variation goes to zero. We had our first taste of such arguments in the simple proposition, Proposition 4.1.2. All such arguments have two parts, a geometric and an analytic. The geometric part studies paths in T~ and their lengths with respect to the new weight function. T h e intent of this is to be able to estimate the new height function on the ring. The geometric argument can be technical and lengthy. The analytic part simply calculates the new area explicitly from the new weight function. In general the analytic part is a straightforward calculation. We will prove a number of propositions first in support of the geometric part of the argument.

4.2.2. PROPOSITION.

Let xET~, and let T~o and T~I denote the ends of T~. Then Idi(x, T~o)§

h i ) -

Hi I < 6/i.

(The reader can perhaps improve on the number

6.)

Proof.

We first prove that

Hi ~ di(x, ~o)§ T~l)§

174 J. W. CANNON

(~1

~

xo Fig. 4

~ o

Choose xoET~o, x l E ~ l , /-approximations

D(xo), D(x), E(x)

and

E(xl)

and arcs so and cq joining

n(xo)

to

D(x)

and

E(x)

to

E(xi)

such t h a t

Li(o~j)=di(x,

R / ) , j = 0 , 1.

(See Figure 3.) If any two/-approximations intersect, we assume their frontiers intersect.

Then the union of Fr D(xo), So,

FrD(x), FrS(x), cq,

and

FrE(xt)

contains a path from T~o to R t of Li-length less than

Let

We now prove that

d~(x,

~ o ) + d i ( x , 7~1) + 4 / i .

di( x, ~o)+di( x,

~ 1 ) ~

Hi+6/i.

E = {y E 7~[di(x, y) <~ 2/i}.

(See Figure 4.) Note that E is

open

(in opposition to what one would expect from

continuous

distance functions). By Propositions 4.1.2 and 4.1.3 there is an Li-minimal p a t h a joining the ends of ~ which is joined to an i-approximation

D(x)

by a p a t h

~=ab, bE~, L~(~)<l/i.

In particular s intersects E so t h a t it contains maximal subarcs

sj=xjyj, xj~7~j, sj

missing E.

The argument is completed as follows. We show first t h a t ao and s l cannot intersect a common shingle, for otherwise s could not be minimal, the portion of s near b in E being too long and unnecessary; it follows that

L~(so)+Li(al)<,.L~(s)=Hi(7~).

We next show that

di(x, 7~j)<Li((~j)+3/i. Our

desired inequality follows.

Suppose t h a t a0 and Sl did intersect a common shingle. Pick an i-approximation D to a point of that shingle. Then s 0 U s t t - ) F r D connects 7~o to ~1 so t h a t

Hi(n) <~ ni (So

UO~l UFr D) <

Li(oto

USl) +

1/i.

(1)

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 175 On the other hand, a contains a subarc 7 = ( b , y ) open-irreducible from b to star(a0 U a l , Si). Let

D(y)

be a n / - a p p r o x i m a t i o n to y. T h e n flU'), forms an i-approximate p a t h from x to y so that

2/i

< Li(~U'~) <

Li(~)+l/i.

(2)

But no shingle intersects b o t h "~ and a 0 U a l . Hence

L,('y) + L i ( a 0 U a l ) ~<

ii(a) = Hi(n).

(3) From (1), (2), and (3) we deduce that Hi < H i , a contradiction.

It remains to show t h a t

di(x,7~j)<Li(aj)+3/i.

Since a j is maximal, there is a point zj of a M E in a shingle which contains

yj.

^Let

D(zj)

be a n / - a p p r o x i m a t i o n to zj,

Dj(x)

a n / - a p p r o x i m a t i o n to x, and ~j a p a t h joining

FrOj(x)

to

FrO(zj)

of ni-length

<.2/i.

Then the union of

~j, FrO(zj),

and a j joins

FrDj(x)

to :Rj and has Li-length

<Li(aj)+ 3/i. []

4.2.3. PROPOSITION.

Suppose r<di(x, 7~j). Then di(n(r), ~;)+r <~ di(x, nj)+ l/i.

Proof.

Recall that

D(r)=D(x, r, i).

We treat two exceptional cases separately:

(1)

D(x)Nn(y)50

for some i-approximations

n(x)

of x and

D(y)

of

y, yEn\D(r).

Then r < di (x, y) <

1/i.

Since

di CO(r), T~j) <~ di (x, 7~j),

we have

di(D(r), ~j)+r < di(x,

^{T~j)+ 1/i}

as desired.

(2)

D(r)MO(z)~O

for some i-approximation

O(z)

of z, z E ~ j . T h e n

di(O(r), T~j)<.

di(D(r),z)<l/i.

Since

r<di(x,n~)

^{w e} have

di(O(r), T~j)+r < 1/i+di(x, T~j)

as desired.

Suppose therefore t h a t neither special case is satisfied. Take a n / - a p p r o x i m a t e p a t h a of minimal length from x to T~j, a joining/-approximations

D(x)

of x and

D(z)

of z, z E n j . Let

~--(a, b)Ca

be open irreducible from

O(z)

to

FrO(r), aED(z), bEFrO(r).

Since we are not in case (2), f~, though possibly degenerate, is not empty. Let

D(r) c

denote the complement of

D(r)

in 7~. Let

~/=(c, d)Ca

be open-irreducible from

D(x)

^to

star O ( r ) c,

ceO(x), dEstarO(r) ~.

^Let

D(y)

i-approximate

yeO(r) ~

in such a way that dET~ Int

D(y).

Note that

D(x)

and

D(y)

are disjoint since we are not in case (1). T h e n ~/

joins

D(x)

and

D(y)

so that

r<di(x, y)<~Li(~/).

Since no shingle intersects b o t h ]3 and %

Li(~)+Li(~/)<.Li(a).

^Since

bEFr(O(r)),

there is a point

weD(r)

and i-approximation

D(w)

such that bET~ I n t O ( w ) . Hence ~ joins

O(w)

to

D(z)

and

di(D(r),T~j)<~Li(~).

Hence

di(O(r), T~j)-Fr< Li(a)--di(x, T~j). []

176 J . w . CANNON

4.2.4. PROPOSITION.

Let 7~o and 7~1 denote the boundary components of T~. Let xETt. Let Hi denote the Qi-height of T~. Then di(x, ~j)<~ Hi+ l/i.

Proof.

By Propositions 4.1.2 and 4.1.3, there is an Li-minimal path c~ joining the ends of T~, an i-approximation

D(x)

to x, and a p a t h / 3 of Li-length <

1/i

joining

D(x)

to c~. T h e n c~U~ joins x to b o t h 7 ~ and n l and has

Li(aU~)<~ii(~)§247

[]

4.2.5. PROPOSITION.

Let ~o and T~I denote the boundary components of T~. Let xET~. Let a be an i-approximate path from x to T~I of minimal Li-length. Then

Li(c~M star(Bd n ) ) <

3/i.

Proof.

If star 2 ~0Mstar 2 7~1 r then Hi

<2/i,

and hence

Li(~) <3/i

by Proposi- tions 4.1.2 and 4.1.3. Therefore we assume the intersection empty. Let yET~I be such t h a t c~ joins i-approximations

D(x)

to x and

D(y)

to y. If

D(x)M(D(y)Ustar 2

T~1)~O, then

D(x)

intersects s o m e / - a p p r o x i m a t i o n to a point of T~I so that Li(c~)<

1/i.

Hence we may assume the intersection empty. We consider two cases.

Case

1: c~Mstar2T~o~O. Pick an arc ~ in c~ open-irreducible from star2R0 to D ( y ) U s t a r ~ 7~1. T h e n

Li(c~M

star(Bd ~ ) ) §

Li (~) <<. Li (~) <<. Hi + 1/i < Li(~) § 3/i.

Thus

Li

(aMstar(Bd ~ ) ) <

3/i as

claimed.

Case

2: aMstar2T~0=O. Pick an arc /3 in a open-irreducible from

D(x)

to D ( y ) U s t a r 2 T~I. T h e n

Li

(c~ M star(Bd ~ ) ) +

Li (/~) <~ Li (~) <~ Li (~) + 1/i.

In this case L i ( a M s t a r ( B d T~)) <

1/i. []

For the remainder of this subsection, we will generally be considering only one i and one x E ~ . Hence we retain the simplified notation,

D(r) = D(x, r, i) = (y e n idi(x, y) <. r).

Despite the closed inequality, ~<,

D(r)

is an open set and not entirely well-behaved. We expand

D(r)

slightly so as to make it a closed and p a t h connected set E ( r ) as follows.

There exist finitely many i-approximations

D1, D2, ..., Dk,

E l , E2, ...,

Ek

T H E C O M B I N A T O R I A L R I E M A N N M A P P I N G T H E O R E M 177 a n d arcs

Oil, 0/2, -.., Ot k

such that:

(i) For each yED(r), there exists a j such that D s is a n / - a p p r o x i m a t i o n to y. For each j, D s is a n / - a p p r o x i m a t i o n to s o m e yED(r).

(2) For each j ,

E s

is a n / - a p p r o x i m a t i o n to x.

(3) If two of

D1, D2, ..., Dk, E l , E2, ..., Ek

intersect, their frontiers intersect in general position. If

D s QE s ~ 0 ,

then a s is a constant path in Fr D s QFr E s. Otherwise a s irreducibly joins Fr Dj to Fr

E s.

(4) For each j , if

(~j(~FrEj,

then

L~(aj)<r.

T h e n

E(r)

is the union of the disks

Dj,

the arcs C~S, and the disks

Ej.

4.2.6. PROPOSITION.

The space E(r) is compact, arcwise connected and satisfies D(r) C E(r) C D(r + l/i).

Proof.

Only the set inclusion

E(r)CD(r+l/i)

needs to be verified. Let

yeE(r).

If

yE~j

for some j , then

yCD(r).

If

yCDjUEj

for some j , then pick an i-approximation

D(y)

for y. We assume t h a t intersecting/-approximations have intersecting frontiers.

If

D(y)

intersects Ej, we find an i-approximate p a t h from y to x in

FrD(y)MFrEj.

If

D(y)

hits

Dj

and Dj hits

Ej,

then we find a n / - a p p r o x i m a t e p a t h from y to x in F r D j . Finally, if neither of these conditions holds, then

Li(aj)<~r

and there is a n / - a p p r o x i m a t e path from y to x in

FrDj[_J~j.

In any case the p a t h described has L c l e n g t h

<.r+l/i. []

We lose no generality in assuming that, for all i and for all xET~, all/-approximations which intersect an i-approximation of x all lie in a single proper disk in T~.

4.2.7. PROPOSITION.

IrE(r) does not lift to the universal cover ~ of T~, then there is a loop C in E(r) which is noncontractible in T~ and has length L~(C)<.2r+4/i.

Proof.

Recall that

E(r)is

the union of finitely many sets of the form

DjuajuEj.

Each of these individually lifts to 7~. This is clear when

DjMEj=O,

for in t h a t case

Dj U~j UEj

is contractible. Otherwise

D j u a s u E j =DjUE s

178 j . W . CANNON

and, by our supposition in the p a r a g r a p h preceding the s t a t e m e n t of the proposition,

Dj UEj

lies in a p r o p e r disk in ~ , hence lifts to ~ . We determine a specific lift of each set

DjUajUEj as

follows. Pick a lift for E l . E v e r y set

E1UEj

has a unique lift extending the lift of E l . T h e n

DjUajUEj

has a unique lift extending the lift of E j .

Now suppose t h a t

E(r)

does not lift. Take a noncontractible simple closed curve J in

E(r).

H o m o t o p J in t u r n out of Int D1, ..., Int

Dk,

Int E l , ..., Int

Ek.

T h e n the h o m o t o p e d J lies in

X = U ( F r D3Ua~U~ E3).

J

Hence X is not llftable. Therefore there is a point z in X and indices j , which we m a y take to be 1 and 2, such t h a t

z E (FrD1UalUFrE1)N(FrD2U(~2UFrE2)

and such t h a t the prescribed lifts Zl and z2 of z defined by lifting

and

D1 Ual UE1

D2Ua2UE2

are different. Pick

x'EFrE1NFrE2.

Let Cj, j = 1 , 2 , be a p a t h from

x'

to z in

Fr DjUajUFr Ej.

T h e n

C=C1.C2

is noncontractible, lies in

Fr DI Ual UPr E1UPr E2Ua2UFr D2 C E(r)

and therefore has length

Li (C) <~ 2r + 4/i. []

We wish to find a similar controlled simple closed curve circling T~ in the case where the set

E(r)

does lift to the universal cover. We spend some time and prove two propositions before we are able to state and prove the desired result, Proposition 4.2.10.

We will make successively greater restrictions on the integer i which we consider.

We will want our simple closed curve to lie in a certain horizontally controlled area B which we now define. We are still assuming t h a t x, i, a n d r are fixed. Let Hi denote the Li-height of ~ . Let

ao=di(x,7~o)

and

al=di(x, 7~l).

Let

bj=max{1/i, a j - l l / i } .

Define

By= {yeT~[d~(y, nj)<~ bj}.

T h e sets B0 and B1 are open. Let B denote the complement of the union of their closures.

Note t h a t the open sets B0 and B1 contain 77.o and ~ 1 , respectively.

THE COMBINATORIAL RIEMANN MAPPING THEOREM 179 4.2.8. PROPOSITION.

If 6/i<Hi, then

star2BoQstar2B1 = ~.

For such i, the open set B contains a simple closed curve which separates the ends of

"R,.

Proof.

If star 2 B0 nstar 2 B1 ~ O, then there exist points Yo E Bo and Yl E B 1 such that any two of their i-approximations

D(yo) and D(yl)

intersect. We may pick xoERo,

x l e n ~ ,

/-approximations

D(xo), D(yo), D(yl), D(xl),

and paths t~j with

Li(aj)<~bj

joining

D(xi)

to

D(yj)

such that if any two of these i-approximations intersect, their

frontiers intersect. Then

Pr D(xo) Uao UPr D(yo) UPr D(yl) U~l UFr D(xl)

joins the ends of 7~. Hence

Hi <~ Li(Fr D(xo ) ) + Li(c~o ) + Li(Fr D(yo ) ) + Li(Fr D(yl ) ) + Li(~I ) + Li(Fr D( xl ) )

< 4 / i + m a x { l l i , (~o- 111i}+max{1/i, al -

11/i}.

The possibilities, up to interchange of 0 and 1, are

Hi <. 6/i,

(1)

Hi <<. ao-6/i,

(2)

Hi ~ ao+al - 18/i.

(3)

The first contradicts our assumption

6/i < Hi.

The second and third conflict with Propo- sition 4.2.2 which states that

ao+al<Hi+6/i.

The closures B-o and B1 are therefore disjoint, closed sets containing T~ and 7~1 and separated by B. By the unicoherence of the 2-sphere 7~/{7~o,T~1}, some component of B separates. And, in a 2-sphere, if an open set separates two points, it contains a simple

closed curve separating those two points. []

4.2.9. PROPOSITION.

Ifr>14/i, then there is an arc aj in BiUD(r ) joining T~j to an i-approximation Dj(x) of x.

COROLLARY.

If r>14/i, then there is an arc oL in BoUD(r)UB1 joining the ends of T"~.

Proof of the corollary.

Note that

FrDj(z)CD(r).

If

F~ Dj(x)

intersects

Rj,

then replace c~j by the empty set. We may assume

Pr Do(x)nFr Dl(x)

r

~.